Use this for solving problem cases, where the limit on the right side exists or is +-infinity. These are problem Cases.
This is L’Hospital rule, used for the problem cases above.
This is how to rearrange your limits so you can use L’Hospital’s rule and still find the limit even though it has an indeterminate form. The situations are: Indeterminate quotient, Indeterminate product, Indeterminate difference and indeterminate powers.
Also, a note on derivatives and the chain rule. This is what you do with constants. Including e, which is a constant.
The Midterm will cover up to section 2.5, and next week the classes will be taught by our TA’s, the professor will be away traveling. The midterm will be heavy on limits.
Limits – Check
A) Be careful, pay attention when f(x) changes rules at a
B) It is hard to compute limits when dividing by 0.
Rules for Limits
Any reasonable operation is allowed, as long as the limit exists and does not divide by 0.
These are mostly easy examples except 1 and 4.
RULE If a division = 0/0 then factor numerator and denominator by (x-a). This set demonstrated this.
The first one here was an easier example of the 0/0 rule. The second one (sin) is REALLY IMPORTANT. And the third one shows how #2 might be used.
This was a challenging problem with sin=1 rule.
This is a neat trick I did not know about for factoring.
For Radicals the trick is to rationalize by multiplying by the Conjugate! The professor stopped here so I better investigate this in the text book.
I am just going to type out the examples gave by the professor.
- Always compute in radians not degrees
- The limit is computed by looking very near the point in question
- Check the right and left side of the limit
- IF x is closs to a, and less than a, f(x) is close L-
- A limit that is always growing bigger is a limit of ∞
Horizontal Line TestA function is one-to-one if and only if no horizontal line intersections it graph more than once.
Domain = independent variable = possible values of x in f(x)
Range = Dependent variable = possible solutions to f(x)
Inverse sine function
Tangent and Velocity
The Slope of the tangent line is the limit of the slopes of the secant lines
This is extra stuff:
A. The limit of f(x) as x approaches a, equals L
B. f(x) approaches L as x approaches a.
Derivatives and Rates of Change